document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=431642>Question 431642</A>: <I><SPAN STYLE=\"word-break: break-all;\"> PROBLEM 1:<BR>\n" );
document.write( "A two digit number with 2 different digits has a special property: \"When the sum of its digits is added to the product of its digits, the result is the number itself.\"  What is the smallest number with this property?<BR>\n" );
document.write( "PROBLEM 2:  <BR>\n" );
document.write( "Six is a perfect number because its factors ( not including 6)add up to itself.  What are all the perfect numbers between 20 and 30?   </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #299581 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=299581\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> 1. Suppose the number is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=10a+%2B+b ALIGN=MIDDLE  ALT=\"10a+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >, where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%3C%3E+b ALIGN=MIDDLE  ALT=\"a+%3C%3E+b\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Then, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+b+%2B+ab+=+10a+%2B+b ALIGN=MIDDLE  ALT=\"a+%2B+b+%2B+ab+=+10a+%2B+b\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a+%2B+ab+=+10a ALIGN=MIDDLE  ALT=\"a+%2B+ab+=+10a\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1+%2B+b+=+10 ALIGN=MIDDLE  ALT=\"1+%2B+b+=+10\"   DISABLED_style= \"max-width:90%; height:auto;\" >, b = 9. We can assume <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a ALIGN=MIDDLE  ALT=\"a\"   DISABLED_style= \"max-width:90%; height:auto;\" > to be as small as possible, so 19 is the smallest such number.\r<BR>\n" );
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document.write( "2. 28, because the sum of its proper divisors is 1+2+4+7+14 = 28. In fact, all even perfect numbers are in the form <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E%28n-1%29%29%282%5En+-+1%29 ALIGN=MIDDLE  ALT=\"%282%5E%28n-1%29%29%282%5En+-+1%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > where <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5En+-+1 ALIGN=MIDDLE  ALT=\"2%5En+-+1\"   DISABLED_style= \"max-width:90%; height:auto;\" > is prime. This is because the sum of divisors function is a multiplicative function for relatively prime integers. When <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=n+=+3 ALIGN=MIDDLE  ALT=\"n+=+3\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=2%5E3+-+1+=+7 ALIGN=MIDDLE  ALT=\"2%5E3+-+1+=+7\"   DISABLED_style= \"max-width:90%; height:auto;\" >, prime, so <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%282%5E2%29%282%5E3+-+1%29+=+28 ALIGN=MIDDLE  ALT=\"%282%5E2%29%282%5E3+-+1%29+=+28\"   DISABLED_style= \"max-width:90%; height:auto;\" >, a perfect number. </SPAN>\n" );
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